DOI 10.1007/s11005-011-0541-6
A Variant of the Mukai Pairing via Deformation
Quantization
AJAY C. RAMADOSS
Departement Mathematik, ETH Z¨urich, R¨amistrasse 101, 8092 Zurich, Switzerland. e-mail: ajay.ramadoss@math.ethz.ch
Received: 21 April 2011 / Revised: 16 November 2011 / Accepted: 17 November 2011 Published online: 30 November 2011 – © Springer 2011
Abstract. Let X be a smooth projective complex variety. The Hochschild homology
HH•(X) of X is an important invariant of X, which is isomorphic to the Hodge cohomol-ogy of X via the Hochschild–Kostant–Rosenberg isomorphism. On HH•(X), one has the Mukai pairing constructed by Caldararu. An explicit formula for the Mukai pairing at the level of Hodge cohomology was proven by the author in an earlier work (following ideas of Markarian). This formula implies a similar explicit formula for a closely related variant of the Mukai pairing on HH•(X). The latter pairing on HH•(X) is intimately linked to the study of Fourier–Mukai transforms of complex projective varieties. We give a new method to prove a formula computing the aforementioned variant of Caldararu’s Mukai pairing. Our method is based on some important results in the area of deformation quantization. In particular, we use part of the work of Kashiwara and Schapira on Deformation Quan-tization modules together with an algebraic index theorem of Bressler, Nest and Tsygan. Our new method explicitly shows that the “Noncommutative Riemann–Roch” implies the classical Riemann–Roch. Further, it is hoped that our method would be useful for gener-alization to settings involving certain singular varieties.
Mathematics Subject Classification (2000). 19L10, 14C40, 19D55, 53D55. Keywords. Mukai pairing, Hochschild homology, periodic cyclic homology,
algebraic index theorem, Euler class, deformation quantization.
1. Introduction
1.1. BACKGROUNDLet X denote a smooth projective complex variety (we remind the reader that X has the Zariski topology). We denote the corresponding (compact) complex man-ifold by Xan. The Hochschild homology HH•(X) is an important algebraic geo-metric invariant of X . The Hochschild–Kostant–Rosenberg (HKR) isomorphism IHKR: HH•(X) → ⊕iHi−•(X, iX) identifies HH•(X) with the Hodge cohomology
complex varieties, then one has a Kunneth isomorphism identifying HH•(X) ⊗ HH•(Y ) with HH•(X × Y ) (see [16,23,27]).
One key reason for the importance of Hochschild homology is its connection to Fourier–Mukai transforms in algebraic geometry. Recall that ifE is an element of the bounded derived category Db(X × Y ) of coherent sheaves on X × Y , then E may be viewed as the kernel of a Fourier–Mukai transform
E: Db(X) → Db(Y ), F → πY∗(E ⊗ πX∗F). (1)
A corresponding Fourier–Mukai transform
HH
E : HH•(X) → HH•(Y )
is defined on Hochschild homologies in several a priori different ways in [5,15,27]. These definitions have since been shown to be equivalent in [23]. In particular, if
E ∈ Db(X), one obtains HH
E : C ∼= HH•(pt) → HH•(X)
by viewingE as the kernel of a Fourier Mukai transform from pt to X. Thus, one can define the Hochschild class
ChHH(E) := HHE (1) ∈ HH0(X)
of E as in [5]. The notation ChHH(E) is used to remind the reader of the close analogy and relation of the Hochschild class with the Chern character. Indeed, Theorem 4.5 of [6] shows that
IHKR(ChHH(E)) = Ch(E) ∈ ⊕iHi(X, iX).
The Hochschild homology HH•(X) has another interesting structure. In [5], Caldararu defined a Mukai pairing −, −M on HH•(X). On the other hand, one
has the Hochschild–Kostant–Rosenberg (HKR) isomorphism IHKR: HH•(X) → ⊕iHi−•(X, iX). The following result was implicitly proven in [18] (and explicitly
so in [21] following [18]). THEOREM 1. a, bM= X IHKR(b) ∧ J(IHKR(a)) ∧ Td(TX).
Here, J is the involution multiplying an element of Hi−•(X, iX) by (−1)i. The Mukai pairing is closely related to another pairing −, −Shk on HH•(X) that was constructed in [23] following Shkyarov in [27]. Different constructions of the same pairing have also appeared in [15,16]. The most transparent definition of −, −Shk is the one given in [16]. This definition states that −, −Shk is given by the com-posite map
HH•(X) ⊗ HH•(X) → HH•(X × X)
HH
O
→ HH•(pt) ∼= C.
Here, O:= ∗OX where : X → X × X denotes the diagonal embedding. The
object O of Db(X × X) is viewed as the kernel of a Fourier–Mukai transform from X× X to pt in the above definition.
A careful comparison between −, −Shk and −, −M was used together with
Theorem 1 in [23] to show the following result. The aim of the current paper is to present a new proof of the same result.
THEOREM 2. a, bShk=
X
IHKR(a) ∧ IHKR(b) ∧ Td(TX). (2)
The usefulness of Theorem 2 stems from the following result about Fourier– Mukai transforms on Hochschild homology (see [16,23,27]).
THEOREM 3. E(x) = x, ChHH(E)Shk ∈ HH•(Y ) for any x ∈ HH•(X).
The right hand side of the formula in Theorem 3 is an abuse of notation. Its correct interpretation is as follows: first, one identifies HH•(X ×Y ) with HH•(X)⊗ HH•(Y ) via the (inverse of the) Kunneth isomorphism. Hence, x ⊗ ChHH(E) is viewed as an element of HH•(X) ⊗ HH•(X) ⊗ HH•(Y ). To this, we apply the homomorphism −, −Shk⊗ idHH•(Y ) : HH•(X) ⊗ HH•(X) ⊗ HH•(Y ) → HH•(Y ) to obtain the R.H.S of the formula in Theorem 3.
Theorem 2 together with Theorem 3 has been of interest in recent years. Not surprisingly, Theorem 1 (equivalently, Theorem 2) implies the Grothendieck Riemann–Roch theorem for smooth projective complex varieties (see [18,19,21]) as well as an explicit version of the Cardy condition (see [24]). Another (pos-sibly more) interesting application of these results is their use for the study of derived equivalences of certain classes of algebraic varieties (for example, K3 sur-faces in [11,17]). We remark that as far as (the above cited as well as other) recent applications are concerned, a formula for −, −Shk is as useful/suitable as one for −, −M (i.e., Theorems 1 and 2 are as useful as one another).
1.2. ABOUT THIS PAPER
In this paper, we provide a different proof of Theorem 2 based on the work of Kashiwara–Schapira [14,15] and an algebraic index theorem of Bressler, Nest and Tsygan in [1–3]. The latter result, one version of which is Theorem 4.6.1 of [2], was proven as part of the authors’ resolution of a conjecture of Schapira and Schneiders (Conjecture 8.5 of [26]) pertaining to the Euler class of D-modules. Unlike the earlier approach from [18,21,23] (also see [25] for further details), this
approach requires that we work overC. However, it gives a clear connection (hith-erto missing) between the computation of a “Mukai pairing” and a large body of work in deformation quantization, algebraic index theorems and related topics. Specifically, our paper shows how the “noncommutative Riemann–Roch” (which is what the Bressler–Nest–Tsygan result amounts to) implies the classical Riemann– Roch.
Our new proof of Theorem2 proceeds as follows. Results from Section 5 of [15] are first used in Section 2 to reduce Theorem 2 to the statement that the Euler class Eu(OX) of the structure sheaf of X coincides with the Todd genus Td(TX)
of the tangent bundle of X . Till recently, this statement was a conjecture of Kashiwara dating back to 1991. In what follows, we shall refer to this statement as Kashiwara’s conjecture.
We prove Kashiwara’s conjecture in Section 3. Our proof uses a proposition (Proposition 5) comparing the Hochschild homology HH•(X) := HH•(OX)
with the Hochschild homology HH•(DXan) of the sheaf of holomorphic
differ-ential operators on Xan. Proposition 5 is proven via some standard arguments from Proposition 3, a similar comparison result for periodic cyclic homologies. Proposition 3 in turn follows from Theorem 4.6.1 of [2]. In order to stick to the main narrative in Section3, we postpone the proof of Proposition3 to Section 4. The argument that Proposition5 implies Kashiwara’s conjecture has three main ingredients. These are: the proof of Proposition 5.2.3 of [15], the Riemann– Roch–Hirzebruch theorem for holomorphic differential operators from [7,22] and Proposition 7. Proposition 7, a technical result computing the Hochschild homol-ogy HH•(Perf(DX)) of the DG-category of perfect right DX-modules, is analogous
to Theorem 5.2 of [12]. Its proof uses generalizations due to Yao (in [32]) of cer-tain deep propositions of Thomason and Trobaugh in [30].
We finish this introduction by pointing out that Kashiwara’s conjecture has been proven in [9] using a deformation to the normal cone argument. While the (inter-esting) approach in [9] is far more concise than the one via [18,21,23], the argu-ment there is geometric and not intrinsic to X . Readers with some background in deformation quantization and algebraic index theory would also find the approach to Theorem 2 in this paper far more concise than the earlier one (that in [18,21,
23]), while remaining algebraic and intrinsic to X in nature. Further, unlike the earlier approach, this method is likely to lend itself to generalization to more gen-eral settings involving certain singular varieties.
2. Preliminaries
Let ωX:= nX[n]. Let : X → X × X denote the diagonal embedding. Recall from
[15] that one has the following commutative diagram in the bounded derived cat-egory Db(OX) of coherent sheaves on X.
∗∗OX −−−−→ td !∗ωX ⏐⏐ IHKR ⏐⏐IHKR ⊕iiX[i] −−−−→ τ ⊕i i X[i]
Here, the map td is as constructed in Section 5.2 of [15] and the map IHKR is as constructed in [15], Section 5.4.1 Let D denote the map on hypercohomology induced by td: ∗∗OX→ !∗ωX. Let IHKR, IHKR and τ continue to denote the maps induced on hypercohomology by IHKR, IHKR and τ, respectively. Applying hypercohomologies, one obtains the following commutative diagram.
H−•(X, ∗∗O X) −−−−→ D H −•(X, !∗ω X) ⏐⏐ IHKR ⏐⏐IHKR ⊕iHi−•(X, iX) −−−−→τ ⊕iHi−•(X, iX) (3)
Kashiwara and Schapira show us in [15]2,3 that
PROPOSITION 1. Theorem2 is equivalent to the assertion that the map τ in (3) is
the wedge product with Td(T X).
Proof. Let X, Y be smooth projective varieties over C. Recall that any ∈
Dbcoh(X × Y ) gives an integral transform cal∗ : HH•(X) → HH•(Y ) (see Section 4.3 of [5]). On hypercohomologies, Corollary 4.2.2 of [15] yields a pairing
−, −KS: HH•(X) ⊗ HH•(X) → C.
We remark that HH•(X) is also the hypercohomology of the complex of Hochs-child chains of OopX , which is equal to HH•(X) since OopX = OX. In particular,
we are not making this identification via the duality map described at the end of Section 4.1 of [15]. Lemma 4.3.4 of [15] then tells us that after identifying HH•(X × Y ) with HH•(Y ) ⊗ HH•(X),4
cal
∗ (α) = Ch(), αKS. (4)
We point out that the right hand side of Equation (4) involves an abuse of nota-tion and that its correct interpretanota-tion is analogous to that of the right hand side of Theorem 3. Let = O ( here denoting the diagonal in X × X). In this case,
1A similar map has been constructed in Section 1 of [21].
2We remark that all constructions/results in Chapter 5 of [15], which are done in the setting of
complex manifolds, work in the algebraic setting that we are working in.
3As mentioned at the beginning of Chapter 5 of [15], most of the work of Chapter 5 in [15]
dates back to a letter [13] written by Kashiwara to Schapira in 1991.
cal
∗ = id (see Section 5 of [5]). Then, by Theorem 5 of [23]5, Ch() =iei⊗ fi,
where the ei and fj are homogenous bases of HH•(X) such that fj, eiShk= δi j.
On the other hand, Equation (4) applied toα =ei tells us that fj, eiKS=δi j, thus
showing that −, −KS= −, −Shk. Finally, the end of Section 5.4 of [15] shows us that
a, bKS=
X
IHKR(a) ∧ τ(IHKR(b)).
We therefore, need to show thatτ =(−∧Td(T X)). In our method, the following proposition from [15], Chapter 5 is the first step in this direction.
PROPOSITION 2. (i) ∗∗OX is a ring object in Db(OX), and !∗ωX is a left module object over ∗∗OX in Db(OX).
(ii) Further, td is a morphism of left ∗∗OX modules in Db(OX).
Proof. The ring structure of∗∗OX in Db(OX) is given by the composite map ∗∗O
X⊗LOX∗∗OX∼= ∗(∗OX⊗LOX×X∗OX)
∗μ
−→ ∗∗O
X,
where μ is induced by the product map ∗OX⊗OX×X∗OX.
The module structure of !∗ωX over ∗∗OX is realized via the composite
map
∗∗OX⊗LOX!∗ωX= ∼ !(∗OX⊗OLX×X∗ωX)
!ν
−→ !∗ωX.
Here, ν is the composite map
∗OX⊗LOX×X∗ωX∼= ∗(∗∗OX⊗OXωX) → ∗ωX the last arrow being induced by the adjunction ∗∗OX→ OX.
The morphism td was constructed in [15] as follows.
∗∗OX∼= OX⊗LOX∗∗OX ∼ = !(OX ωX) ⊗LOX∗∗OX ∼ = !((OX ωX) ⊗LOX×X∗OX) !((OX ωX) ⊗LOX×X∗OX) ∼= !∗ωX
That td is a morphism of left ∗∗OX-modules is more or less a direct
conse-quence of the fact that ⊗LO
X is associative.
5Note that we are not using any part of [23] that depends on the Mukai pairing formula
COROLLARY 1. For all α ∈ ⊕iHi−•(X, iX), τ(α) = α ∧ τ(1).
Proof. The ring structure of∗∗OX induces a product• on H−•(X, ∗∗OX).
By Proposition 2, D(a • b) = a • D(b)
for all a, b ∈ H−•(X, ∗∗OX). It follows from Lemma 5.4.7 of [15] that for all
a, b ∈ H−•(X, ∗∗OX),
IHKR(IHKR(a) ∧ β) = a • IHKR(β).
The desired corollary now follows from the fact that IHKR and IHKR are isomor-phisms.
Recall that for any E∈ Db(OX), one has the Chern character ch(E) ∈ H0(X, ∗ ∗OX). By Theorem 4.5 of [6], IHKR(ch(E)) is the Chern character of E in the
classical sense. The Euler class Eu(E) is defined as the element IHKR −1
(D(ch((E)))
of ⊕iHi(X, iX). Note τ(1) = Eu(OX). In order to compute the −, −Shk, we therefore, need to show that
Eu(OX) = Td(TX).
Before we proceed, let us make a clarification. Recall that ∗∗OX is
repre-sented in the derived category D−(OX) of bounded above complexes of
quasi-coherent sheaves on X by the complex of C•(OX) of completed Hochschild chains
(after turning it into a cochain complex by inverting degrees). Recall from [33] that
Cn(OX):=lim←− k
O⊗n+1X Ik
n , where In is the kernel of the product map O ⊗n+1
X →OX. Let C•(OX) be the complex of sheaves of X associated with the complex of presheaves
U→ C•( (U, OX)) (the Hochschild chain complex here being the naive algebraic
one). One similarly defines C•red(OX) using reduced Hochschild chains. There are
natural maps C•red(OX) ← C•(OX) → C•(OX) of complexes of sheaves on X which
are quasi-isomorphisms. In the following section, when thinking of the complex of Hochschild chains on X , we shall be thinking of C•red(OX) (which has the same
hypercohomology as C•(OX)).
3. The Euler Class of
O
XIt remains to show that Eu(OX) = Td(TX). This fact was originally conjectured
(in 1991) by Kashiwara in [13]. The original intrinsic computation proving this from [18] (see [21] for details) is very lengthy and involved. Further, its connec-tions to deformation quantization and related areas are not clear. Another, more recent proof due to [9] uses deformation to the normal cone. We now sketch
our new approach to this question. Let DX denote the sheaf of (algebraic)
dif-ferential operators on X . Recall that the Hochschild–Kostant–Rosenberg quasi-isomorphism on Hochschild chains induces an quasi-isomorphism IHKR: HCper0 (OX) →
∞
p=−∞H2 p(Xan, C). On the other hand, a construction very similar to the trace
density construction of Engeli–Felder on Hochschild chains induces an isomor-phism χ : HCper0 (DXan) →∞
p=−∞H2n−2p(Xan, C) (see [7,20,28]). Further, one has
a natural map (−)an: HCper0 (DX) → HCper0 (DXan)6. The natural homomorphism
OX→DX of sheaves of algebras on X induces maps on Hochschild as well as
neg-ative cyclic and periodic cyclic homologies. These maps shall be denoted byι. The following proposition is closely related to a Theorem in [1] (also see [2,3]). It will be proved in Section 4.
PROPOSITION 3. The following diagram commutes: HCper0 (OX) −−−−→ (−)an◦ι HC per 0 (DXan) ⏐⏐ IHKR χ⏐⏐ ∞ p=−∞H2 p(Xan, C) (−∧Td(TX)) −−−−−−→ ∞p=−∞H2n−2p(Xan, C)
Note that for any sheaf of algebras A on X, one has natural maps HC−0(A) → HCper0 (A) and HC−0(A) → HH0(A). Also recall that one has a natural projection H2 p(Xan, C) → Hp,p(Xan, C) for all p.
PROPOSITION 4. The following diagrams commute: (a) HC−0(OX) IHKR −−−−→ ∞p=−∞H2 p(Xan, C) ⏐⏐ ⏐⏐ HH0(OX) IHKR −−−−→ ⊕pHp,p(Xan, C) (b) HC−0(DXan) −−−−→χ ∞ p=−∞H2n−2p(Xan, C) ⏐⏐ ⏐⏐ HH0(DXan) −−−−→χ H2n(Xan, C)
6Indeed, if f : Xan→ X is the canonical map, one has a natural map f−1(CCper • (DX)) →
CCper• (DXan) of complexes of sheaves on Xan, and hence in the derived category D(ShC(Xan)) of sheaves of C-vector spaces on Xan. By adjunction, one gets a natural map CCper• (DX) →
R f∗(CCper• (DXan)), to which we apply R (X, −). R f∗ and R extend to D(ShC(Xan)) and D(ShC(X)), respectively, since f∗ and (X, −) have finite cohomological dimension.
(c) HC−0(OX) (−) an◦ι −−−−→ HC−0(DXan) ⏐⏐ ⏐⏐ HH0(OX) (−) an◦ι −−−−→ HH0(DXan)
Proof. We prove part (a), leaving the proof of the remaining parts to the reader.
Let Cp denote the complex
0→ pX−→ dDR pX+1−→ · · ·dDR
of sheaves (of C-vector spaces) on X with pX in (cohomological) degree 0. There is a natural map Cp→ pX of complexes of sheaves on X given by the identity on Xp (here, Xp is thought of as a complex concentrated in degree 0). One also has a natural map of complexes Cp→•
X,DR[p] of sheaves on X, where •X,DR[p]
denotes the algebraic De-Rham complex of X with a shift. The following diagram clearly commutes: HC−0(OX) IHKR −−−−→ H0(X, ⊕ pCp[p]) ⏐⏐ ⏐⏐ HH0(OX) IHKR −−−−→ H0(X, ⊕ ppX[p])
It therefore, suffices to show that the natural map H0(X, Cp[p]) → Hp(X, p X)
coincides with the composite map7 H0(X, Cp[p]) → H0(X, •X,DR[2p]) ∼= H2 p
(Xan, C) → Hp,p(Xan, C) after one identifies Hp(X, p
X) with H
p,p(Xan, C). Note that the hypercohomology H0(X, Cp[p]) may be computed by passing to Xan and replacing each iX by the corresponding Dolbeault resolution to obtain a double complex, the 0th cohomology of whose total complex is H0(X, Cp[p]). Hence, any
class inα ∈H0(X, Cp[p]) is represented by a harmonic 2p-form ω :=⊕pr2 pωr,2p−r
on Xan, where ω
r,2p−r is a harmonic (r, 2p − r)-form on Xan. The image of α
in H2 p(Xan, C) is also represented by ω. Clearly, the image of α in Hp(X, pX) is represented by ωp,p, which coincides with the projection from H2 p(Xan, C) to
Hp,p(Xan, C) applied to the class of ω in H2 p(Xan, C).
7Indeed, the composition of direct sum (over p) of the composite maps H0(X, Cp[p]) →
H0(X, •
X,DR[2p]) ∼= H2 p(Xan, C) with IHKR: HC−0(OX) → H0(X, ⊕pCp[p]) is what we denote by
PROPOSITION 5. The following diagram commutes: HH0(OX) (−) an◦ι −−−−→ HH0(DXan) ⏐⏐ IHKR χ⏐⏐ ⊕pHp,p(Xan, C) −−−−−−−−→ (−∧Td(TX))2n H2n(Xan, C)
Proof. We note that the natural map HC−0(OX)→HH0(OX) is surjective. Indeed,
after applying IHKR, we are reduced to verifying that Hp(X, Ker(d :Xp→ p+1
X ))→
Hp(X, Xp) is surjective. By Serre’s GAGA, it suffices to verify that Hp(Xan, Ker(d :
p
Xan→ pX+1an)) → Hp(Xan, pXan) is surjective. This follows from the fact that any
closed (p, p)-form defines an element of Hp(Xan, Ker(d : pXan→
p+1
Xan)) as well.
Hence, any y∈ HH0(OX) lifts to an element ˜y ∈ HC−0(OX). For notational
brev-ity, we denoteχ ◦ (−)an by χ for the rest of this proof. Now, χ ◦ ι(y) = (χ ◦ ι( ˜y))2n by Proposition4, parts (b) and (c). Further,(χ ◦ι( ˜y))2n=(IHKR( ˜y)∧Td(TX))2n by Proposition 3. Finally, (IHKR( ˜y) ∧ Td(TX))2n= (IHKR(y) ∧ Td(TX))2n by Proposi-tion 3, part (a) and the fact that Td(TX) ∈ ⊕pHp,p(Xan, C).
The following proposition is a crucial point in this note. PROPOSITION 6. The following diagram commutes:
HH0(OX) D −−−−→ H0(X, ! ∗ωX) ⏐⏐ (−)an◦ι ⏐⏐ ( IHKR−1(−))2n HH0(DXan) −−−−→ Hχ 2n(Xan, C)
Proof. Letπ : X → pt be the natural projection. The object OX of Perf(OX×pt)
induces a morphism π∗: Perf(OX) → Perf(pt) in the homotopy category
Ho(dg-cat) of DG-categories modulo quasi-equivalences (see Section 8 of [29]). The nota-tion π∗ is justified by the fact that the functor from D(Perf(X)) to D(Perf(pt)) induced by π∗ is indeed the derived pushforward π∗. This induces a map π∗: HH0(OX) → HH0(Opt) = C which coincides with the pushforward on Hochschild
homologies from [15] (see Theorem 5 of [23]). On the other hand, one has π∗: ⊕pHp,p(Xan, C)→H0(pt, C)=C, which coincides with
Xan. By the proof of
Prop-osition 5.2.3 of [15], IHKR −1
◦ D commutes with π∗. On the other hand, let Perf(DX)
denote the DG-category of perfect complexes of (right)DX-modules that are
quasi-coherent asOX-modules. One has a map π∗D: Perf(DX) → Perf(pt) in Ho(dg-cat).
The functor induced by π∗D on derived categories maps M ∈ D(Perf(DX)) to π∗(Man ⊗LDX an OX
an).8 By Section 8 of [29], π∗D induces a map π∗D : HH0
(Perf(DX))→HH0(pt)∼=C on Hochschild homologies. By Proposition7at the end of this section, the composite map
HH•(Perf(DX)) → HH•(DX)(−)
an
→ HH•(DXan) (5)
is an isomorphism (the first map in the above composition is the trace map from Section 4 of [12]). π∗D therefore, induces a C-linear functional on HH0(DXan),
which we shall continue to denote by π∗D. It follows from Theorem 1.1 of [7] and Corollary 1 of [22] that
π∗D=
Xan
◦χ : HH0(DXan) → C.
Since Xan: H2n(Xan, C) → C is an isomorphism, the required proposition follows
once we check that π∗=π∗D◦(−)an◦ι. This follows from the fact that the diagram HH0(OX) (−) an◦ι −−−−→ HH0(DXan) ⏐ ⏐ ⏐⏐ HH0(Perf(OX)) (−)⊗OXDX −−−−−−→ HH0(Perf(DX))
(the left vertical arrow being the trace isomorphism from Section 4 of [12] and the right vertical arrow being the composite map (5)) commutes as well as the obser-vation that for E∈ D(Perf(OX)),
π∗Dι(E) = π∗((E ⊗OXDX) an⊗L
DX anOX
an) = π∗Ean
(recall that π∗E= π∗Ean in D(Perf(pt)) by Serre’s GAGA). Let α ∈ HH0(OX) be arbitrary. By Proposition 6,
IHKR−1(D(α))2n= χ(ι(α)an). By Proposition 5,
χ(ι(α)an) = (I
HKR(α) ∧ Td(TX))2n.
By Corollary 1, IHKR−1(D(α)) = IHKR(α) ∧ Eu(OX). Hence, (IHKR(α) ∧ Eu(OX))2n= (IHKR(α) ∧ Td(TX))2n
for all α ∈ HH0(OX). Because wedge-and-integrate is a perfect pairing, Eu(OX) = τ(1)=Td(TX). To complete the proof of Proposition 6, we sketch the proof of the
following proposition.
PROPOSITION 7. HH•(Perf(DX)) ∼= HH•(DXan). This isomorphism is realized by
Proof. One has to verify that the arguments of Keller in Section 5 of [12] go through when OX is replaced by DX. The crucial part here is the analog of
The-orem 5.5 of [12] (originally proven as Propositions 5.2.2–5.2.4 of [30]) whenOX is
replaced by DX. This is done in Propositions 3.3.1–3.3.3 of [32] (which prove the
analog of Theorem 5.5 of [12] in a much more general setting: in particular, when
OX is replaced byRX, whereRX is a sheaf of quasi-coherent OX-algebras
(possi-bly noncommutative)). Let Y be any quasi-compact, quasi-separated scheme over C with V ,W quasi-compact open subschemes of Y such that Y = V ∪ W. Following the arguments of Sections 5.6 and 5.7 of [12], one obtains a morphism of Mayer– Vietoris sequences
HHi(Perf(DY)) −−−−−−−→ HHi(Perf(DV)) ⊕ HHi(Perf(DW)) −−−−−−−→ HHi(Perf(DV∩W)) −−−−−−−→ HHi−1(Perf(DY)) ⏐⏐
⏐⏐ ⏐⏐ ⏐⏐
HHi(DYan) −−−−−−−→ HHi(DVan) ⊕ HHi(DWan) −−−−−−−→ HHi(D(V ∩W)an) −−−−−−−→ HHi−1(DYan)
(for each i∈Z). The vertical arrows in the above diagram are induced by the com-posite map (5). As in Section 5.9 of [12], we may then reduce the proof of the desired proposition to proving the desired proposition when X is affine with trivial tangent bundle. For the rest of this proof, we assume that this is indeed the case. LetDX-mod denote the Abelian category of (right)DX-modules are
quasi-coher-ent OX-modules. There is an equivalence of abelian categories between DX-mod
and DX-mod, where DX:= (X, DX) (see [32], example 1.1.5). Hence, one has an
equivalence of DG-categories between Perf(DX) and Perf(DX) (this follows, for
instance, from Lemma 2.2.1 of [32]). This equivalence induces an isomorphism HH•(Perf(DX))
∼ =
→ HH•(Perf(DX)). Further, there is a natural map HH•(DX) →
HH•(DX) such that the following diagram commutes:
HH•(Perf(DX)) ∼ = −−−−→ HH•(Perf(DX)) ⏐⏐ ∼= ⏐⏐ HH•(DX) −−−−→ HH•(DX)
In the above diagram, the vertical arrows are trace maps from Section 4 of [12]. For honest algebras, they yield isomorphisms. We are therefore, reduced to verify-ing that the composite map
HH•(DX) → HH•(DX)(−)
an
→ HH•(DXan) (6)
is an isomorphism. Let D•Xan denote the Dolbeault resolution of the sheaf DXan.
This is a sheaf of DG-algebras on X . Let C•(D•Xan) denote the complex of global
sections of the complex of completed Hochschild chains on X (see [22], Section 3.3). There is a natural map of complexes C•(DX) → C•(D•Xan) inducing (6) on
homology. To prove that this is a quasi-isomorphism, we filter algebraic and holomorphic differential operators by order and consider the induced map on the E2-terms of the spectral sequences from Section 3.3 of [4]. This turns out to be
induced on homology by the natural map from the algebraic De-Rham complex
(2n−•(T∗X), dalg
DR) to the Dolbeault complex ( (Xan, 2n−•
T∗Xan⊗OX an0X,•an), d + ¯∂).9
That this is a quasi-isomorphism amounts to the assertion that the natural map from the algebraic De-Rham complex of X to the smooth De-Rham complex of Xan is a quasi-isomorphism (see [10]).
4. A Proof of Proposition
3
One notes that the following diagram commutes: HCper0 (OX) (−) an −−−−→ HCper 0 (OXan) ⏐⏐ ι ι⏐⏐ HCper0 (DX) (−) an −−−−→ HCper 0 (DXan)
To prove Proposition 3, it therefore, suffices to show that the following diagram commutes (where Y:= Xan): HCper0 (OY) −−−−→ ι HC per 0 (DY) ⏐⏐ IHKR χ⏐⏐ ∞ p=−∞H2 p(Y, C) (−∧Td(TY)) −−−−−−→ ∞p=−∞H2n−2p(Y, C) (7)
In other words, we now work with a complex manifold rather than an algebraic variety. Recall that there is a deformation quantization AT∗Y of OT∗Y[[]] such
that π−1DY→ AT∗Y[−1] and AT∗Y[−1] are flat over π−1DY. Here, π : T∗Y→ Y
is the canonical projection.
In this situation, one has a natural map π−1: HCper0 (DY) → HCper0 (AT∗Y[−1]).
Indeed, ifU :={Ui} is a good open cover of Y , one has a natural map of complexes
between the periodic cyclic-Cech complex C∨(U, CCper• (DY)) and C∨(V, CCper• (A
T∗Y[−1])), where V := {π−1(Ui)}. Similarly, one has a natural map π−1: HCper0 (OY) → HCper0,C(AT∗Y)10. Further, one has a trace density map χFFS: HC
per 0 (AT∗Y
[−1]) →
pH2n−2p(T∗Y, C)(()) (see [1,7,8,28]). Note that we can compose χFFS with the natural mapβ :HCper0,C(AT∗Y)→HC
per
0 (AT∗Y[−1]).11 We shall abuse
nota-tion to denote χFFS◦ β by χFFS. Let i: Y → T∗Y denote inclusion as the zero sec-tion. The following proposition is clear.
9Here, •
T∗Xan is the complex of sheaves on Xan whose sections on each open subset U of
Xan are holomorphic forms on T∗U that are algebraic along the fibres of the projection T∗U→U. d is the (holomorphic) De-Rham differential on this complex.
10The subscript C here means that the tensor product used in defining Hochschild, and hence,
periodic cyclic chains is over C.
11β is the composite map HCper
0,C(AT∗Y) → HC
per
0,C(AT∗Y[−1]) → HC
per
PROPOSITION 8. The diagram HCper0 (DY) π −1 −−−−→ HCper0 (AT∗Y[−1]) ⏐⏐ χ χFFS⏐⏐ pH2n−2p(Y, C)(()) π∗ −−−−→ pH2n−2p(T∗Y, C)(()) commutes. Further, i∗◦ π∗= id on pH2n−2p(Y, C)(()).
One has a “principal symbol” homomorphism σ : AT∗Y→ OT∗Y. The following
theorem is from [1]. The reader may also refer to [2,3] and section 7 of [31] in this context. The particular statement we want is immediate from a statement in Section 1.2.7 of [2]. The latter statement is a consequence of Theorem 4.6.1 of [2], as explained in the proof of Theorem 3.3.1 of [2].
THEOREM 4. The following diagram commutes:
HCper0,C(AT∗Y) σ −−−−→ HCper0 (OT∗Y) ⏐⏐ χFFS IHKR⏐⏐ pH2n−2p(T∗Y, C)(()) (−)∪π∗Td(TY) ←−−−−−−−− pH2 p(T∗Y, C)(())
The following proposition is clear as well.
PROPOSITION 9. The following diagrams commute:
HCper0 (OY) π −1 −−−−→ HCper 0,C(AT∗Y) ⏐⏐ π∗ id⏐⏐ HCper0 (OT∗Y) ←−−−− HCσ per0,C(AT∗Y) HCper0 (OY) −−−−→ι HCper0 (DY) ⏐⏐ π−1 π−1⏐⏐ HCper0,C(AT∗Y) β −−−−→ HCper 0 (AT∗Y[−1]) HCper0 (OY) π ∗ −−−−→ HCper0 (OT∗Y) ⏐⏐ IHKR IHKR⏐⏐ pH2 p(Y, C) π∗ −−−−→ pH2 p(T∗Y, C)(())
Denote the bottom arrow in the diagram of Equation (7) by θ (after extending scalars to C(()) in the codomain). Let α ∈ HCper0 (OY) be arbitrary and let β :=
IHKR(α). Then, θ(β) = χ(ι(α)) by definition of θ = i∗(π∗(χ(ι(α)))) = i∗(χFFS(π−1(ι(α)))) by Proposition 8 = i∗(χFFS(π−1(α))) by Proposition 9 = i∗(I HKR(σ(π−1(α))) ∪ π∗(Td(TX))) by Theorem 4 = i∗(I HKR(σ(π−1(α)))) ∪ Td(TX) = i∗(IHKR(π∗(α))) ∪ Td(TX) by Proposition 9 = i∗(π∗(β)) ∪ Td(TX) by Proposition 9 = β ∪ Td(TX).
Since IHKR: HCper0 (OY) →pH2 p(Y, C) is an isomorphism, Proposition 3 follows
from the above computation.
Acknowledgements
I am especially grateful to Xiang Tang and Boris Tsygan for very useful discus-sions (in particular, for helping me clarify a question I had about Theorem 4.6.1 from [2]). In fact, this note is motivated by a joint project with Xiang Tang. I am also very grateful to Damien Calaque, Giovanni Felder and Pierre Schapira for some very useful discussions. I also convey my heartfelt thanks to both the refer-ees for their suggestions, which played an important role in improving the presen-tation of this paper. This work is partially supported by the Swiss National Science Foundation under the project “Topological quantum mechanics and index theo-rems” (Ambizione Beitrag Nr. PZ00P2 127427/1).
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